#Polymath project full#

Find the smallest unit-distance graph of a given minimum degree (excluding, in some natural way, boring cases like Cartesian products of a graph with a hypercube).Р”Р‡ollowing guidance from Terry Tao, we are setting a deadline for this project at Apr 15, 2019.Improve the current record of 105/29 for the fractional chromatic number of the plane.Improve the corresponding bound in higher dimensions.Find a 6-chromatic unit-distance graph.Goal 3: Apply these simpler graphs to inform progress in related areas.

#Polymath project full#

This has so far been decreased to 278, though the corresponding full graph is actually larger than AubreyРІР‚в„ўs original one. Computer assistance was leveraged in to analyze a subgraph of size 397. Goal 2: Reduce (ideally to zero) the reliance on computer assistance for the proof.

The current record is 1577, and recent insights promise to lead to further progress soon. Goal 1: Find progressively smaller 5-chromatic unit-distance graphs. Along these lines, our goals for this project are quite varied: This has led to interest in the search for simpler examples. However, the smallest such graph that he discussed has 1581 vertices, and its lack of a 4-colouring requires checking for the nonexistence of particular types of 4-colourings of subgraphs of it that have almost 400 vertices, which requires a computer search. In April 2018, Aubrey posted to the arXiv a report of a family of 5-chromatic unit-distance graphs. It is easy to find 4-chromatic unit-distance graphs, the smallest being the 7-vertex Moser spindle. Such graphs are termed unit-distance graphs. It was first posed in 1950 and was reduced the following year to a finite graph problem in particular, CNP is at least k if and only if there exists a finite graph of chromatic number k that can be drawn in the plane with each edge being a straight line of unit length. The HadwigerРІР‚вЂњNelson problem is that of determining the chromatic number of the plane (CNP), defined as the minimum number of colours that must be assigned to the points of the plane in order to prevent any two points unit distance apart from being the same colour. We will summarize progress on the Polymath wiki page. Discussion of the project of a non-research nature should continue in the Polymath proposal page. This project is a follow-up to AubreyРІР‚в„ўs breakthrough result that the chromatic number of the plane is at least 5. Given the initial activity on the Polymath proposal page, Terry Tao suggested that Aubrey de Grey and I launch Polymath16 to make progress on the HadwigerРІР‚вЂњNelson problem.